Standard conjectures for abelian fourfolds

• 成果类型: Article
• 署名作者: Ancona, Giuseppe
• 署名单位: Universites de Strasbourg Etablissements Associes; Universite de Strasbourg
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00990-7
• 发表日期: 2021
• 页码: 149-212
• 关键词: numerical equivalence; algebraic cycles; VARIETIES; lefschetz; DECOMPOSITION; CONSTRUCTION; motives

摘要

Let A be an abelian fourfold in characteristic p. We prove the standard conjecture of Hodge type for A, namely that the intersection product Z(num)(2)(A)(Q) x Z(num)(2) (A)(Q) -> Q is of signature (rho(2) - rho(1) + 1; rho(1) - 1), with rho(n) = dim Z(num)(n)(A)(Q). (Equivalently, it is positive definite when restricted to primitive classes for any choice of the polarization.) The approach consists in reformulating this question into a p-adic problem and then using p-adic Hodge theory to solve it. By combining this result with a theorem of Clozel we deduce that numerical equivalence on A coincides with l-adic homological equivalence on A for infinitely many prime numbers l. Hence, what is missing among the standard conjectures for abelian fourfolds is l-independency of l-adic homological equivalence.